2023 NJC P1 Q3

It is given that \(y={{\left( 1+{{\tan }^{-1}}x \right)}^{2}}\).

1. Prove that \({{\left( 1+{{x}^{2}} \right)}^{2}}\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+2x\left( 1+{{x}^{2}} \right)\frac{\mathrm{d}y}{\mathrm{d}x}=2\).[3]
2. Obtain an equation relating \(\frac{{{\mathrm{d}}^{3}}y}{\mathrm{d}{{x}^{3}}}\), \(\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}\) and \(\frac{\mathrm{d}y}{\mathrm{d}x}\).[1]
3. Find the Maclaurin’s series for \(y\), up to and including the terms in \({{x}^{3}}\).[2]
4. Deduce the equation of the normal to the curve \(y={{\left( 1+{{\tan }^{-1}}x \right)}^{2}}\) at \(x=0\).[1]